Thanks everyone. I think I solved it.
I spoke to Samsung who said it probably WASN'T Macrovision, but the DVD Player's resolution wasn't high enough and I should try to boost it. Unfortunately there is no way to do that on this player...or so I thought.
I found the Video EQ Menu and chose the Cinematic setting (listed as the "Best Picture Quality"). That sounds higher resolution to me, so I selected it. Sure enough it works! I have to select it every time I put in a new DVD (I think the player remembers each disc's setting) but it seems to work pretty well; I've watched whole episodes of The Simpsons and Arrested Development plus a few scenes from The Incredibles and no dropped signal.
As this is all new to me, I won't be surprised if something changes over the next few days. So I may return with more questions.
Special Thanks to "wmcclain" for that cool tip on how to turn off Macrovision and Region Coding. I'll make use of that soon enough.